This study investigated the covariational reasoning abilities of prospective middle school mathematics teachers in a task about dynamic functional events involving two simultaneously changing quantities in an individual process and also in a peer interaction process. The focus was the ways in which prospective teachers' covariational reasoning abilities re-emerge in the peer interaction process in excess of their covariational reasoning. The data sources were taken from the individual written responses of prospective teachers, transcripts of individual comments, and transcripts of conversations in pairs. The data were analyzed for prospective teachers in terms of the cognitive and interactive aspects of individual behavior and also interaction. The findings revealed that prospective teachers at different levels working in pairs benefited from the process in terms of developing an awareness of their own individual and also a pair's understanding of covarying quantities. Furthermore, the prospective teachers had opportunities to develop their knowledge on the connection between variables, rate of change, and slope. The prospective teachers' work in pairs provided salient explanations for their reasoning about the task superior to their individual responses.
As a part of a long-term design-based research project, this study aimed to investigate the mean differences among the pre-, post-and retention tests of seventh grade students regarding their proportional reasoning skills. The data of this paper were collected from 30 seventh-grade students enrolled in a public school in Ankara, Turkey in the first and second semester of 2017-2018 academic year. Participants of the study were engaged in a teaching experiment aimed to develop their meaningful understanding of ratio and proportion unit through authentic activities. The intervention prolonged 30 lesson hours and Proportional Reasoning Test was applied to the students at the beginning of the instruction as pre-test, at the end of the instruction as post-test, and 5 months later as retention test. A one-way repeated measures of ANOVA was conducted to compare scores on the Proportional Reasoning Achievement Test at Time 1 (pretest), Time 2 (posttest) and Time 3 (retention test). Findings revealed that there was a significant difference among proportional reasoning skills. More specifically, there was a statistically significant increase in Proportional Reasoning Achievement Test scores from Time 1 to Time 2 but the difference between Time 2 and Time 3 was not statistically significant. Findings revealed the significant effect of intervention on students' proportional reasoning skills and retention of those skills. Based on the findings it could be deduced that designing authentic interventions could have an impact on enhancing and enduring students' proportional reasoning skills.
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